The Singularity Structure Of Quantum-Mechanical Potentials

We investigate nonrelativistic quantum-mechanical potentials between fermions generated by various classes of quantum field theory (QFT) operators and evaluate their singularity structure. These potentials can be generated either by four-fermion operators or by the exchange of a scalar or vector mediator coupled via renormalizable or nonrenormalizable operators. In the nonrelativistic regime, solving the Schrodinger equation with these potentials provides an accurate description of the scattering process. This procedure requires providing a set of boundary conditions. We first recapitulate the procedure for setting the boundary conditions by matching the first Born approximation in quantum mechanics to the tree-level QFT approximation. Using this procedure, we show that the potentials are nonsingular, despite the presence of terms proportional to r(-3) and del(i)del(j)delta(3)((r) over right arrow). This surprising feature leads us to bifurcate the space of nonrelativistic quantum mechanical potentials into those which can be UV completed to a QFT and those which cannot. We identify preliminary criteria for distinguishing between these two classes of potentials. We also consider extensions to potentials in higher dimensions and find that Coulomb potentials are nonsingular in an arbitrary number of spacetime dimensions.